OpenAI Researchers on the Future of Mathematical Reasoning

In this episode of the A16Z Podcast, host Leisha Li interviews OpenAI mathematicians Mark Selke and Metab Swani about how large language models are advancing mathematical research. The discussion begins with how AI differs from human mathematicians: while humans often abandon approaches after hours or weeks of unsuccessful attempts, AI models persist doggedly without the psychological burden of sunk costs. The researchers describe a moment of breakthrough when GPT-5 rapidly identified whether an Erdős problem was still unsolved by finding relevant literature—a task that had frustrated them for hours. They then discovered 10 similar cases where AI provided crucial connections across fields.

The conversation shifts to recent accomplishments announced through Astra, OpenAI's mathematical reasoning system. The first major result concerns sphere packing in high dimensions. The researchers explain that while sphere packing in 2D (hexagonal lattice) and 3D (grocery store oranges) have solutions, the general problem remains largely mysterious. The optimal packings are only known in five dimensions (1, 2, 3, 8, and 24), with dimensions 8 and 24 corresponding to special mathematical structures (E8 and Leach lattices) that emerged from other areas of mathematics. Previous bounds on sphere packing density in high dimensions, established in the 1970s, used ugly combinatorial optimization arguments and yielded the unintuitive number 2^(-0.599d).

Astra's breakthrough was analyzing the linear programming (LP) bound for sphere packing, a framework developed by Cohn and Elkies. The model was asked to understand how this bound behaves in large dimensions and discovered an elegant asymptotic formula: the bound equals 2^(-(2π + o(1))d), approximately 2^(-0.601d). Crucially, the model proved this is the optimal bound achievable through LP methods—not just a better guess, but a complete characterization. The proof uses complex analysis and is surprisingly short. The researchers emphasize this represents genuine mathematical reasoning: the model identified the right direction to pursue among many possibilities, much like a human mathematician recognizing a promising approach.

The second result concerns spherical codes and binary error-correcting codes, which are dual problems to sphere packing. These are highly relevant to practical communication systems. The model improved bounds for both cases using representation theory—leveraging the symmetry properties of spheres and hypercubes. Interestingly, this was one case where interaction occurred: the researchers asked the model to improve code bounds, then prompted it to push further. The model's more sophisticated representation theory approach ultimately recovered the conjectured value for the original sphere packing problem, revealing a deep connection between the problems. The researchers speculate this interaction may become unnecessary as models improve.

The third major result addresses SOFIC groups, an abstract algebra question. A group is SOFIC if it can be approximated by finite groups—analogous to how the integers can be approximated by integers modulo n. For decades, mathematicians hoped every group was SOFIC, reasoning that finite approximations should work universally. This relates to the famous Aldous-Lyons conjecture about approximating graphs. A counterexample disproving Aldous-Lyons required 250 pages building on prior 200-page work, involving quantum complexity theory. However, Astra produced a direct 15-page proof that non-SOFIC groups exist using only group theory. The proof identifies a delicate combinatorial obstruction that previous researchers had implicitly identified but couldn't rule out. By adding one algebraic fact, the model prevented the "conspiracy" that previous authors couldn't eliminate.

The researchers discuss key differences between AI and human mathematics. Humans often become cognitively locked into failed approaches, unable to properly restart exploration in a new direction without their "context window" being polluted by prior failures. AI can easily spawn parallel sessions exploring different paths. Additionally, AI excels at executing proofs once a general direction is identified—getting epsilon smaller than delta, maintaining algebraic precision—details where humans easily get lost. The researchers note AI seems better at updating how promising a path is versus simply rejecting it, and can make good decisions about which approaches to pursue based on combining existing knowledge with mathematical judgment.

On the question of AI producing genuinely mathematical reasoning versus sophisticated guessing, the researchers present evidence from reasoning traces. The chains of thought read surprisingly like notes from human mathematicians—not alien logic but recognizable mathematical thinking. This emerges from training reasoning models to think deeply, not from domain-specific mathematical training tricks. The researchers note that mathematical proofs are surprisingly short and elegant, contrary to fears of thousand-page incomprehensible AI solutions.

The conversation explores how mathematics will change as proving results becomes less of a bottleneck. Previously, the person proving a result also gained deep understanding and bore responsibility for explaining it. With AI handling proofs more readily, different skills become valuable: understanding results deeply enough to communicate them to others, maintaining communal knowledge, recognizing which problems matter, and demonstrating mathematical taste in directing research. The researchers suggest this could democratize mathematics—people without full-time mathematics careers could understand deep results by using AI to absorb complex papers. They emphasize that even exponential AI improvement won't trivialize everything: hard unsolved problems like P versus NP should remain challenging indefinitely.

The researchers also discuss the practical impact on working mathematicians. One finds it much faster to input a paper PDF and have a model explain the proof strategy than to read carefully. This addresses the problem that AI creates: the explosion of results is simultaneously solved by AI's ability to help absorb mathematics. The mathematical community has begun building on these results—notably, researchers Kuhn and Tom (whose prior work Astra built upon) have already published follow-up work exploring more examples of non-SOFIC groups. The researchers express hope that the community will continue this pattern rather than viewing AI results as endpoint achievements.